The microrheology of colloidal dispersions

The Microrheology of Colloidal Dispersions XlI. Trajectories of Orthokinetic Pair-Collisions of Latex Spheres in a Simple Electrolyte 1

K. TAKAMURA, H. L. GOLDSMITH, AND S. G. MASON Pulp and Paper Research Institute of Canada and Department of Chemistry, McGill University, Montreal, Canada Received August 11, 1980; accepted December 9, 1980 Suspensions of 2.6- and 4.0-/zm-diameter polystyrene latex spheres in aqueous glycerol containing KC1 were subjected to Poiseuille flow in 200-/zm-diameter tubes and the trajectories of particles undergoing two-body collisions recorded on cine film. The results were analyzed using theoretical equations describing the translational and rotational motions of interacting charged spheres in a shear flow. From the asymmetry of the paths of approach and recession of the spheres forming a transient doublet, it was possible to detect net attractive or repulsive forces of the order of 10-13 N acting for 0.1 to 0.35 sec while the surfaces approached closer than 100 nm. The interaction forces were interpreted using the DLVO theory of colloid stability. Thus, in 50% aqueous glycerol, at 1 m M KC1, net repulsive forces due to double layer interaction acted during the collision, while at 10 mM KC1, the existence of a net attractive force provided evidence that the spheres passed through the secondary potential energy minimum. Here, trajectory analysis yielded a value of the Hamaker constant of 0.003 aJ and the retardation parameter of the van der Waals force of 200 nm. In 93% glycerol, however, double layer interaction did not appear to be involved in the pronounced repulsion observed during some collisions. It may be that this effect was due to surface roughness or the presence of hydration forces physically preventing approach of the two surfaces closer than i0 nm. INTRODUCTION

This paper is concerned with two-body (orthokinetic) collisions between charged, equal-sized spheres of colloidal dimensions in aqueous suspensions subjected to simple shear flow at low Reynolds numbers. The trajectory equations describing the translational and rotational motions of the spheres in terms of the hydrodynamic and interparticle forces acting along the particle centers have recently been developed (1). In an earlier paper (2), these equations were used to describe the shear-induced rotation of permanent doublets of 2.6-/zm-diameter polystyrene (PS) latex spheres about the vorticity axis of the flow. Measurements of i This work was supported by Grant MT-4012 of the Medical Research Council of Canada.

the mean dimensionless period of rotation TG (T is the period of rotation, G is the shear rate) at various KC1 concentrations indicated the existence of two kinds of doublet, one with the spheres in the secondary energy minimum (separation distance h between surfaces > 17 nm), and the other with the spheres in the primary energy minimum (h being much smaller). From the shift in distribution in TG with KC1 concentration it was shown that the Hamaker constant A, of the PS latex lay between 0.006 and 0.01 aJ. In the present work, the trajectory equations are used to describe the collision paths of transient doublets whose spheres separate after colliding. Such paths have been extensively investigated for doublets of macroscopic rigid spheres in viscous fluids undergoing Couette or Poiseuille flow when the

175 0021-9797/81/070175-15502.00/0 Journal of Colloid and Interface Science, Vol. 82, No. 1, July 1981

Copyright © 1981 by Academic Press, Inc. All fights of reproduction in any form reserved.

176

TAKAMURA, GOLDSMITH, AND MASON

The results described below demonstrate particle motions are governed by purely hydrodynamic forces (3,4). With the exception that marked effects of interparticle forces, of collisions occurring near the vessel wall, repulsive or attractive depending on the it was shown that the paths of approach electrolyte concentration, on the sphere traand recession of the spheres were curvilinear jectories can be observed. and mirror images of one another. It should be noted that by collision we do not imply E X P E R I M E N T A L PART that the particles actually touch each other; indeed, theory shows (5) that a layer of fluid 1. Apparatus. The experiments were perremains between the surfaces and that the formed in a traveling microtube apparatus spheres continue to rotate during their en- (2, 15, 16) using -0.01% v/v latex suspencounter. Nevertheless, the sphere surfaces sions flowing through 200-/~m-diameter precan come very close to each other with the cision bore glass tubes at volume flow rates minimum theoretical separation distance, Q - 30/~1 hr -1, as described in the previous hmin, as small as the dimensions of a single study (2). Particle collisions were observed atom, especially when the encounter occurs and tracked at 500× magnification through between spheres of colloidal size and in the a fixed, high-resolution microscope focused equatorial plane of the shear field. When the on the median, X2X3-plane of the tube norparticles are charged and of colloidal dimen- mal to the viewing, Xl-axis (Fig. 1). To sions there are significant effects of inter- achieve high resolution of the image of the particle forces, Fint(h), due to double layer colliding spheres, and to record events ocrepulsion and van der Waals attraction on curring in <5 sec at sufficiently low exposure their trajectories. Depending upon the exact times (<6 msec), a 16-mm Locam cine camform of Fint(h) the paths of approach and era having a 1/9 shutter (Red Lake Labs, recession are no longer symmetrical and the Santa Clara, CA), operating at framing trajectories of recession are the cumulative speeds from 20 to 30 sec -1 was used, inresult of the forces acting during collision. stead of the TV-system described in the preIn principle, therefore, it is possible to es- vious study. 2. Materials. Suspensions of 2.6-/xm-ditimate the magnitude and sign of Fint(h) by measuring the detailed trajectories of the ameter surfactant-free PS latex were prespheres of a transient doublet in shear flow. pared as described in the previous paper (2). The present study describes such experi- The 4.0-/xm-diameter PS latex, used in other ments using 2.6- and 4.0-/zm-diameter PS studies reported from this laboratory (16), spheres in dilute solutions of KC1 in aqueous was kindly provided by Prof. R. H. Ottewill. glycerol. Stored aqueous suspensions of these latices To date few direct measurements of van were diluted to 0.01% by volume with 50 der Waals attractive or double layer repul- or 93% aqueous glycerol containing the sive forces have been attempted. Such force desired KC1 concentration. The glycerol measurements were made either in macro- served to raise the viscosity of the suspendscopic systems using mica plates (6-8) or ing medium to - 5 and 250 mPa s, remetal wires (9), or with dispersions of highly spectively, thereby considerably reducing concentrated latex or clay (10-12). By Brownian diffusion of the spheres and the contrast, the present work focuses on the perturbation of the particle trajectories interactions between two colloidal-size which would have interfered with the measparticles in extremely dilute aqueous suspen- urements of Fint(h). sions, i.e., under conditions when the The electrophoretic mobility U of the latex DLVO theory of colloid stability may be spheres was measured in a microelectroexpected to apply (13, 14). phoresis apparatus (Rank Brothers, CamJournal of Collold and Interface Science, Vol. 82, No, 1, July 1981

MICRORHEOLOGY OF COLLOIDAL DISPERSIONS bridge, England). g-potentials were then calculated f r o m U = Keo (],1o, K and "00being the respective dielectric constant and viscosity of the suspending medium, and e0 the permittivity of free space. N o relaxation correction was n e c e s s a r y since in these suspensions Kb > 100 (K is the reciprocal Deb y e thickness and b the sphere radius). The g-potentials of both latex spheres in 1, 10, and 100 m M KC1 are s u m m a r i z e d in Table I. The reasons for the o b s e r v e d differences

-U3(R)

177

TABLE I ~-Potentials of the Polystyrene Latices g-potential (mV) in [KCI] mM

Water

50% glycerol

2.6 Nm

1 10 100

-40 -17 -12

-44 -45 -10

4.0 ~m

1

-15

(-5) a

Latex

Value estimated from trajectories of colliding spheres (Fig. 8) since the electrophoretic mobilities were too low to measure accurately,

-U3(R)

between the values in water and glycerol solutions are not understood. 3. Procedures. The particle encounters chosen for filming during approach, collision, and separation were those in which both spheres a p p e a r e d to be in the s a m e focal plane and traveled with v e r y low relative velocity, thus resulting in angles of a p p r o a c h tb~ > 50 ° (Fig. 1) n e c e s s a r y to achieve close distances of a p p r o a c h b e t w e e n sphere surfaces. Because of the depth of focus ( - ---I /xm) only about 10% of the encounters photographed actually resulted in transient doublets having both spheres apparently in the XzX3-plane, and it was mainly these that were subsequently analyzed, frame by frame on a projection table using a stop-motion projector (Model M-16C, Vanguard InstruL ment Corp., Melville, NY). The shear rate G(R) at the center of the doublet axis was obtained f r o m the particle radial position R in the tube and the velocity distribution U3(R), the latter being separately m e a s u r e d at the beginning and end of each e x p e r i m e n t by photographing flow with the tube at rest (2). The ~b~-orientation of X3 the doublet axis was obtained b y using the FIG. 1. Schematic diagram of a two-body collision tube wall as a reference line. between rigid spheres which form a transient doublet. Shown is the median plane of a tube of radius R0 being RESULTS moved upward with the velocity U3(R) of the downward flowing fluid at the mid-point of the axis between l. General Observations the two spheres at a radial distance R from the tube The experiments were carried out with center. Cartesian (X1, X2, X3), and polar coordinates 2.6and 4.0-/xm PS latex spheres at KC1 con(01, 4~) with X1 as the polar axis (also the viewing axis) are constructed at the mid-pointof the doublet axis. centrations of l, 10, and 100 raM, the first Journal of Colloid and Interface Science, Vol. 82, No. 1, July 1981

178

TAKAMURA, GOLDSMITH, AND MASON

(°}

X2

f [b)

(cJ

X2~~k

X2~

Attraction

DoubletFormotion

FIG. 2. Trajectories of shear-induced collisions of 2.6-/zm PS latex spheres in 50% aqueous glycerol showing the projection on the X2X3-plane of the paths of the centers of spheres from the midpoint between them. At the center is the exclusion sphere which cannot be penetrated if the collision occurs in the X~X3-plane. (a) 4.0-/.tm-diameter spheres in 1 mM KCI: a significant increase in the distance of separation of centers along the X2-axis from 0.15b before, to - 0 . 8 b after collision has occurred; Fint(h) > 0. (b) 2.6-p.m-diameter spheres in 10 m M KCI: the paths of approach and recession are almost symmetrical; actually Fint(h) < 0 as shown by subsequent analysis (Fig. 6a). (c) 4.0-/zm spheres in 100 mM KCI: the collision resulted in the formation of a permanent doublet. The asterisks denote the point at which there was interaction with a third sphere. The interactions shown here were almost equatorial (01 ~ 90°). lournal of Colloid and Interface Science,

Vol. 82, No. 1, July 1981

corresponding to the condition for a stable suspension, and the latter two for the formation of secondary and primary doublets, respectively (2). More than 50 encounters were observed at each KC1 concentration. In 100 mM KC1, the suspension was quite unstable with the formation of many large aggregates which occasionally blocked the entrance of the tube, necessitating frequent cleaning and changing of the suspension during the experiments. By shortening the duration of each experiment it was possible to film and analyze the formation of permanent doublets. By contrast, the latex suspensions were stable at the lower KCI concentration. In 10 mM KC1 the formation of permanent, presumably secondary doublets was observed, although very infrequently, and no cine films of such encounters were obtained. With the exception of a doublet of short lifetime described below, the formation of permanent doublets was not observed in 1 mM KC1. Here, as in 10 mM KC1, the experiments focused on the effects of Fint on the trajectories of transient doublets. For most collisions, visual observations revealed no obvious asymmetry of the paths of approach and recession, so it was left to a detailed analysis of cine films of the trajectories and ~bl-orientations of the spheres of 20 transient doublets to determine the effects of the interaction forces.

2. Trajectories Typical trajectories of transient doublets in 1 and 10 mM KC1, respectively, are shown in Figs. 2a and b, and that of the formation of a permanent doublet in 100 mM KC1 in Fig. 2c. The two curves in each figure represent the projection on the X2X3-plane of the paths of the centers of the two spheres of the doublet, r' apart, from the midpoint between them as a function of the angle ~bl. At the center is the exclusion sphere which cannot be penetrated by either colliding particle if the trajectories are truly in the equatorial, X2X3-plane.

MICRORHEOLOGY OF COLLOIDAL DISPERSIONS X2

FIG. 3. Trajectory, as in Fig. 2 of a collision between 4-/zm PS latex spheres in aqueous 1 mM KCI showing the formation of a doublet having a lifetime of 1 orbit. Unlike the collisions in Fig. 2, this is a nonequatorial encounter (0, < 90°) with the projection of the paths inside the exclusion sphere. The marked scatter of points is due to translational Brownian motion; G = 1.5 sec-1; closed circles, approach; open circles, recession. Figure 2a illustrates the effect of a net repulsive force on the trajectories. Here, the spheres initially approached along paths havingx2 = ___0.15b while moving parallel to the X3-axis, but receded along a path having x2 = _+0.8b. Whereas the paths of approach became curvilinear at r' -~ 1.3b, ~b~ = - 8 0 ° and the separation distance between surfaces reached the limit of resolution of the microscope at ~bl = - 4 5 °, the sphere surfaces visually separated already at ~bl = +10 °. This was accompanied by a marked increase in the relative translational velocities of the spheres after collision. By contrast, the trajectories at 10 m M KC1 were nearly symmetrical, the paths of approach and recession being almost mirror images of each other (Fig. 2b), although detailed analysis of the trajectory described below, revealed the presence of a weak attractive force. In Fig. 2c, a p e r m a n e n t doublet was formed during the encounter between the spheres. The decrease in x2 at r' = 6b during the path of approach (at the point indicated by the asterisk) was due to an interaction with a third sphere. Also, the collision occurred close to the wall of the tube where optical distortion introduced errors in the measurement of the particle positions and these may account for the points which lie outside the boundary of the exclusion

179

sphere. After the formation of a stable orbit pair, the doublet was observed for >20 rotations without separating. In the absence of glycerol, p e r m a n e n t doublets having a short life time of 1 or 2 orbits were observed at all three KC1 concentrations. As illustrated in Fig. 3, these were formed during nonequatorial collisions when 01 < 30 ° at ~bl = 0 and the projection of the paths in the X2X3-plane lay within the exclusion sphere. Here, the scatter in the experimental points was due, in part, to the uncertainty in the measurement of r' when the spheres overlapped and only one could be kept in focus. ANALYSIS OF TRAJECTORIES 1. Theoretical Considerations

In the presence of interparticle forces, Fint(h), the trajectories of two equal sized spheres in shear flow are given by (1, 2): dr*

sin 2 01 sin 2~b~

-A(r*)

dt*

+ dO1 dt* d61 dt*

1 -

-

4 1 2

C(r*)Fint(h)

3~r'0oGb 2

[IA]

B(r*) sin 201 sin 261

[IB]

[1 + B(r*) cos 26,].

[1C]

Here, 0, and 6, are spherical polar coordinates with respect to the polar axis X, (vorticity axis of the flow) with origin at the midpoint of the line of sphere centers, r apart (Fig. 1), t is the time, and t* = Gt. A(r*), B(r*) and C(r*) are known dimensionless functions o f r* (=r/b) which have been given (1, 5, 17). When Fint(h) = 0, the trajectories of approach and recession of the colliding spheres are symmetrical about 6, = 0, and are defined by two constants, D and E, given by the integrated form of Eqs. [1] (5, 17): 1 xl = + - r*Df(r*) 2

[2]

Journal of Colloid and Interface Science, Vol. 82, No. 1, July 1981

180

TAKAMURA, GOLDSMITH, AND MASON [

\

|

I

I

- - I

cule, assuming that the theory applies at such small gap widths. When Fint(h ) :~ 0, the paths of approach and recession are no longer symmetrical; D and E become functions of time, increasing when Fint(h) > 0 (repulsion) and decreasing when Fint(h) < 0 (attraction).

o2~-

[KCI], mM

o_

y O 0 40

20

I 60

2. Fitting the E x p e r i m e n t a l Trajectories i

80

h, nm FiG. 4. Theoretical plot of Fint(h) against h for PS latex spheres in 50% glycerol as computed from Eqs. [5] and [6] using A = 0.006 aJ, h = 100 nm, and 00 assumed = measured ~-potential. Curves 1 and 2 : 4 /zm spheres in 1 m M KCI, 00 = - 4 . 0 mV assuming constant surface charge density and constant surface potential respectively. Curve 3 : 2 . 6 / ~ m spheres in 10 m M KC1; the secondary potential energy minimum (Fint(h) = 0) occurs at h = 35 nm. Curve 4 : 4 /~m spheres in 100 mM KC1; Fint(h) < 0 for allh considered.

x2 = +- 1 r * f ( r * ) [ E + g(r*)] 1/2

2

[31

wheref(r*) and g(r*) are integral function of r*, for which numerical values have been given (5, 17). When D = 0, the collision occurs in the equatorial, X~X3-plane. There exist two types of trajectory: (i) closed trajectories (E < 0) in which the spheres make a hydrodynamically stable orbit pair with the particles in periodic motion about each other; (ii) open trajectories (E > 0) in which the spheres encounter each other once and subsequently separate. E = 0 defines the limiting trajectory separating open from closed trajectories. The minimum separation distance r'in at 61 = 0 varies with the trajectory and can be calculated from the condition (4): f2(r*min)[D2 + E + g(r*in)] = 1.

[4]

It is smallest for equatorial encounters, e.g., for the 2.6- and 4.0-/zm-diameter latex spheres used in the present investigation, when D, E = 0, the minimum separation distance between surfaces, hmin = 4.2 x 10-Sb, less than the size of a water moleJournal of Colloid and Interface Science, Vol. 82, No. 1, July 1981

In principle, Fint(h) can be calculated from Eq. [1A] by inserting the experimentally obtained translational velocity of the sphere centers. However, this was not possible in the present experiments since r* was known only in the case of equatorial collisions (in the median plane of the tube) which constituted less than 1% of all observed collisions. Even then it was impossible to measure dr*/dt* when h < 100 nm, just when interparticle forces begin to be important. There being no practically feasible method of measuring dr*/dt*, recourse was had to fitting the experimentally observed trajectories by numerically integrating Eqs. [1] using expressions for Fint(h) from the DLVO theory of colloidal stability (13, 14): Fint(h ) = Fattr(h ) + Frep(h)

where the van der Waals attractive Fattr(h) and double layer repulsive forces Frep(h) are given by (18, 19): A b [ 1 + 3.54p ] 12h2L( 1 + 1.77p)2] ,

Fattr(h) =

p
A b [ 0.98 1-'~'2 p

+

[5AI 0.434 p2

0.0674 ] ~ 3 ' p > 1

( e--~--~h / Frep(h) = 27rKbeoO~ 1 +- e -'h ] "

[5BI

[6]

Here, A is the Hamaker constant and p = 2rrh/h the retardation parameter, h being the London wavelength. The + and - in Eq.

M I C R O R H E O L O G Y OF C O L L O I D A L D I S P E R S I O N S

~ IO / ~

I

~,°-s,5ol o,:s2°:;° I

2 0/ ~)

6

01

I I

I

-5

I

0'=82°

I

~

0

I I

~ ='81"5°+'0"5°

I

5

f* FIG. 5. Trajectory of the 2.6-/xm spheres of doublet 8 of Table II plotted as the dimensionless projected length r* sin 01 against t* = Gt, in order to c o m p u t e the trajectory c o n s t a n t s D and E. The points are experimental and the lines drawn were obtained by numerically integrating Eq. [1] at large h (Fint(h) = 0) both for approach and recession. T h e best fit was obtained using initial values r* = 6, 491 = - 8 1 . 5 °, and 01 = 82.0 ° corresponding to D = 0.83 and E = 0.77. (a) Effect on the c o m p u t e d trajectory of varying 01 by _+1°. (b) Effect of varying 491 by ___0.5°.

[6] correspond to constant surface potential

tOo and constant surface charge density of the spheres, where tOo is now the surface potential at h = oo. Results of calculations ofFint(h ) vs h for the 2.6- and 4.0-/xm spheres at the three [KC1] are shown in Fig. 4. The analysis of the trajectories was carried out in two stages. First, the paths of approach and recession of the spheres at large h (>200 nm, Fint(h) ~- 0), plotted as r* vs t*, were fitted by numerically integrating Eqs. [1], then solving for D and E using Eqs. [2] and [3], respectively. For the recession trajectory, the integration was performed in reverse time. Since most collisions were nonequatorial, the observed interparticle distances r'/b are the projected lengths = r* sin 01, and in order to integrate Eqs. [1] the initial value of 01 (and hence D) had to be known. The value chosen was that which gave a best fit of the experimental trajectory. Figure 5a shows that varying 01 by _+1° results in no measurable change of the trajectory at large h; it only affects the

181

trajectory when t* ---> 0. By contrast, varying the initial value of ~bl by only _0.5 ° results in a significant change in dr*/dt* during approach and recession (Fig. 5b). With this trial and error method, the trajectory constants D and E could be determined to within _+3 and _+1.5%, respectively. For those collisions in which the trajectory constants increased or decreased from approach to recession, the second stage of analysis involved numerically integrating the complete Eqs. [1] at h < 200 nm using Eqs. [5] and [6] for Fire(h), adjusting the values of the Hamaker constant at a given retardation parameter to give values of Fint(h) which would fit the trajectory of recession at large h. The effect of Fint(h) is thus to increase (net repulsion) or decrease (net attraction) the values ofh during the period of close approach of sphere surfaces and thereby to decrease or increase dr*/dt* and the trajectory constants during recession. 3. Results of the Analysis a. Transient doublets: h >>1/•. The trajectory constants computed for 8 encounters between 2.6-/xm-diameter PS latex spheres in 50% glycerol at 10 mM KC1 are listed in Table II. The Table also gives values ofhmin calculated from Eq. [4], i.e., assuming Fint(h ) -- 0. It is evident that both trajectory constants decreased after collision for those doublets having hrnin < 100 nm, indicating the existence of a net attractive force, Fint(h ) < 0. The trajectories of these doublets were further analyzed to yield values of the Hamaker constant, as follows. In solving Eqs. [5] and [6] there are three unknown variables, tO0,A, and h, and Figs. 6 a - d show the effect on the trajectories of varying one of these while keeping the other two constant. Using the experimentally measured g-potential of -45 mV, assumed = tOo, and ~ = 100 nm, the recession paths calculated for doublet 8 of Table II with A varying from 0.004 to 0.010 aJ are shown in Journal of Colloid and Interface Science, Vol. 82, No. 1, July 1981

182

TAKAMURA, GOLDSMITH, AND MASON T A B L E II Shear-Induced Collisions of 2.6-/.~m-Diameter PS L a t e x Spheres: Calculated Values of Trajectory and H a m a k e r C o n s t a n t s ; [KC1] = 10 m M , 50% Glycerol Trajectory constants Doublet

G (sec-1)

1

6.0

1.9 1.9

0.12 0.12

330

2

4.6

1.4 1.4

0.68 0.65

210

3

5.4

1.3 1.3

0.38 0.36

120

4

4.4

1.1 1.1

0.96 0.91

100

5

6.4

1.2

0.45

80

--

0.001

1.2

0.41

Da

approach recession

Hamaker constant, Ac (aJ) Ea

h rainb

(nm)

~, = 100 nm

h=

6

4.7

0.93 0.90

0.85 0.75

60

0.006

0.001

7

4.6

0.52 0.50

1.4 1.3

55

0.006

0.001

8

5.5

0.83 0.78

0.77 0.59

35

0.006

0.002

a Calculated from the best fit of the trajectories of approach and recession at large h, using Eqs. [2] and [3], respectively. b Calculated from the trajectory of approach using Eq. [4]. e Values estimated from the best fit of the whole trajectory using Eqs. [1] with Fint(h) given by Eqs. [5] and [6] a s s u m i n g ~00 = m e a s u r e d C-potential.

Fig. 6a. The best fit of the measured trajectory was obtained when A =- 0.006 aJ. A similar result was obtained for doublets 6 and 7. As illustrated in Fig. 6b, the trajectories of recession were not v e r y sensitive to changes in ~b0,as might be expected from the fact that h was large compared to the double layer thickness (1/K = 2.8 nm). H o w e v e r , if the value qJ0 = - 17.2 mV found for the latex in aqueous 10 rriM KC1 was used, dr*/dt* during recession was considerably lower. The third variable to be determined is the degree of retardation of the van der Waals force. Generally, h has been assumed to lie between 100 and 200 nm (18, 20-23). The Journal of Colloid and Interface Science, Vol. 82, No. 1, July 1981

trajectories of transient doublets were calculated using values from h = 50 nm (strong retardation) to h = ~ (zero retardation) and, as may be seen by comparing Figs. 6a and 6c and from Table II, reasonable values of A existed over the whole range of h considered. Figure 7 shows the decrease in A with increasing k for doublets 7 (Fig. 6d) and 8; the points for the two doublets lay on a single curve. To fix the respective values of A and h it was necessary to obtain collision trajectories at two different F~ep(h) but at the same Fattr(h). This was done by analyzing collisions in 50% glycerol, 10 m M KCI to which 3/xg d1-1 Cat-floc, a cationic polyelectrolyte

MICRORHEOLOGY OF COLLOIDAL DISPERSIONS - -

I(a) 8~

(b)

PDoubtetS

'~o = -45 mV ' /~ = lOOnm

-2V /" AxlO J

I__

I

J

I

I

I

Doublet 8

,~= 1OOnm A = 6x10-21j

__

I

i

i

(c}

Doublet 8

__

~

183 ~

i

I

J

~ ~ -45 mV

~

r

Doublet 7 -55

~ ,mY •

I/J0= -45mV )~= oo

A~10-21j.

7.2 -

6

t*

Flo. 6. Plots, as in Fig. 5 of the trajectories of the spheres of doublets 7 and 8 of Table II. The points are experimental and here the lines drawn were obtained by numerically integrating Eq. [1] with Fint(h) computed from Eqs. [5] and [6], varying the parameters qJo,A, and h. (a), (d): Effect of varyingA on the trajectories of doublets 8 and 7 of Table II, respectively; qJ0 = -45 mV, h = 100 nm. (b) Effect of varying t~0, doublet 8. (c) Effect of varying A with X = c¢, doublet 8.

(2) h a d b e e n a d d e d . A s will b e s h o w n f r o m a d e t a i l e d a n a l y s i s o f t h e t r a j e c t o r i e s g i v e n in t h e s u c c e e d i n g p a p e r (24), t h e o n l y e f f e c t o f t h e p o l y m e r at s u c h a l o w c o n c e n t r a t i o n , w h e n it w a s l i k e l y i r r e v e r s i b l y a n d flatly ads o r b e d o n t o t h e s u r f a c e , w a s to d e c r e a s e Frep(h) b y i n c r e a s i n g tOo f r o m - 4 5 to - 4 1 inV. T h e r e s u l t o f t h e t r a j e c t o r y a n a l y s i s is s h o w n in Fig. 7 w h e r e t h e p o i n t s n o w lie on a new curve of A vs h which crosses that o b t a i n e d in t h e a b s e n c e o f C a t - f l o c at t h e p o i n t X = 200 n m , A = 0.003 aJ. T h e s e then, are the most suitable values for the retardation parameter and the Hamaker c o n s t a n t o f p o l y s t y r e n e in 50% w / w g l y c e r o l . T h a t a v a l u e o f A = 0.003 aJ in 50% glycerol is r e a s o n a b l e w h e n c o m p a r e d to t h a t , 0.006 aJ < A < 0.01 aJ, p r e v i o u s l y o b t a i n e d in w a t e r c a n b e s h o w n b y a p p l y i n g an e q u a tion g i v e n b y V i n c e n t (25) r e l a t i n g t h e H a m a k e r c o n s t a n t , Am, o f a m i x e d s o l v e n t to t h a t o f t h e p u r e c o m p o n e n t s , All a n d A22, in vacuo: A m = [ V l ( A l l ) 1/2 - (1 - V1)(A22)1/2] 2,

[7]

w h e r e V1 is t h e v o l u m e f r a c t i o n o f c o m p o n e n t 1. U s i n g v a l u e s o f A l l andA22 f o r w a t e r a n d g l y c e r o l o f 0.037 a n d 0.074 aJ, r e s p e c t i v e l y , as o b t a i n e d f r o m r e f r a c t i v e i n d e x m e a s u r e m e n t s at v a r i o u s w a v e l e n g t h s (25, 26), Eq. [7] y i e l d s A m = 0.052 aJ f o r Va = 0.56 in 50% g l y c e r o l w/w. T h e H a m a k e r

0.010

0

200

400

~k , 13m

Flo. 7. Relationship between A and X obtained from the best fit of the computed trajectories of 2.6-/.~m spheres in 50% glycerol, 10 mM KC1; ©, &: doublets 8 and 7, respectively; 0: doublet 8 in 50% glycerol containing 3.0 ~g dl -I Cat-floc. The two best-fit curves drawn through the experimental points intersect at A = 0.003 aJ and X = 200 nm. Journal of Colloid and Interface Science, Vol. 82, No. 1, July 1981

184

TAKAMURA, GOLDSMITH, AND MASON TABLE III

Shear-Induced Collisions of 4.0-/zm-Diameter PS Latex Spheres: Calculated Trajectory Constants; [KCI] = 1 mM, 50% Glycerol Trajectory constants

Doublet

G (sec -t)

Da

approach recession

Eb

hm~.b (nm)

~0c (mY)

9

2.7

0.93 1.0

0.09 0.23

13 33

-4.0

10

2.4

0.88 0.97

0.21 0.38

15 38

-5.0

11

2.0

0.83 0.99

0.04 0.32

8 34

-4.0

12

1.5

0 0

0.08 34

-4.0

-0.004 1.29

,.b Calculated as in Table II. e Estimated from the best fit of the whole trajectory using Eqs. [1] with Fint(h) given by Eqs. [5] and [6] assuming A = 0.003 a.l and h = 200 nm.

constant for polystyrene in this solvent, A3m3, can then be obtained from the relation (20): A3rn3 :

[(A33)1/z - (Am)l/2] 2

[8]

where the Hamaker constant for polystyrene A33 = 0.078 aJ (25). Equation [8] gives a value of 0.0027 aJ for the H a m a k e r constant polystyrene in 50% glycerol, in good agreement with that computed from the collision trajectories. This value, and that of h = 200 nm are used in the remaining trajectory analyses given below. b. T r a n s i e n t d o u b l e t s : h = 1/K. The trajectory constants obtained by fitting the experimental curves of4.0-/zm PS latex spheres in 1 m M KC1 are listed in Table III. H e r e , hmi n w a s calculated both from the path of approach and recession. It is evident that, were Fint(h ) : 0, the spheres would have approached to within h < 15 nm, a distance comparable to the double layer thickness 1/K = 8.8 nm. In fact, hmin as determined from the recession trajectory was of the order of 35 nm in all four cases, suggesting that strong repulsive forces existed at h lower Journal of Colloid and Interface Science, Vol. 82, No. 1, July 1981

than this value. This is also illustrated by the plot of r* sin 01 vs t* for doublet 12 of Table III in Fig. 8. Here, E < 0 for the path of approach, i.e., a stable orbit pair would have formed ifFint(h) = 0 with theory predicting that the sphere surfaces would approach as close as h < 0.1 nm, i.e., less than the atomic dimensions. Clearly, even the smallest surface irregularities would cause a change in trajectory. The actual hmin, as determined from the recession path, was 34 nm. It was difficult to obtain an accurate value of the ~-potential for the 4-/zm PS latex in 50% glycerol because of the very low electrophoretic mobilities, although it was known to be > - 10 mV. Accordingly, the measured recession paths were fitted using h = 200 nm and A = 0.003 aJ by systematically varying tk0. As illustrated in Fig. 8, the best fit was obtained with ~b0 = - 4 . 0 mV; similar values were found for the other doublets of Table III. Since theory shows that, when h is of" the order of l/K, there are significant differences in Fint(h) depending on whether the assumptions of constant surface charge density or constant surface potential are used (Fig. 4, curves 1 and 2), the trajectory cal-

6

Doublet 12 (~o,mV -10

2~-

-~

-5

........

0 t*

-3.5

5

FIG. 8. Effect of ~b0 on the trajectories of doublets of 4.0-p.m spheres in 50% glycerol, 1 mM KC1. The points are experimental and the lines were calculated by numerically integrating Eq. [1] with Fint(h) computed from Eqs. [5] and [6], assuming constant surface charge density. The plot shows that, as experimentally found, the suspension is stable until ~bo > - 3 . 5 mV when permanent doublets are predicted to form.

185

M I C R O R H E O L O G Y OF C O L L O I D A L D I S P E R S I O N S .

i .

i ,

i

culations were carried out for both cases. It was found that, for tOo < - 5 mV, the paths of recession were almost identical, but at '.~ 4 4'0 too = - 4 . 2 mV the constant potential theory predicted that a permanent doublet would form whereas the constant charge theory 10 20 30 40 predicts separation of the particles until too t* = - 3 . 5 mV (Fig. 8). The fact that doublet FIG. 9. Trajectory of the doublet previously s h o w n formation was not observed with the 4-/xm in Fig. 3; plot of r* sin 01 vs t* and, inset, xl vs t*. spheres at this KC1 concentration indicates The points are experimental; the solid lines during apthat the assumption of constant charge denproach and final recession were computed from Eq. sity may be the correct one to describe the [1] with Fint(h) = 0. The values of xl = r* cos 0, were calculated using 0, obtained from the numerical indouble layer interaction. tegration of Eq. [1]. c. Doublet formation. The trajectory constants of the paths of approach of the spheres shown in Fig. 2c were calculated to beD = 0 and E = 0.56, and according to Eq. [4] the applying the DLVO theory to collisions in spheres would separate after approaching to which the sphere surfaces approach to within within h = 3 nm. However, since Fi,t(h) h - 30 rim. To test whether the theory ap< 0 for all h in this range (Fig. 4, curve 4) plies at lower h, trajectory measurements the spheres were captured in the primary were carried out with 4.0-/zm-diameter PS energy minimum. This was confirmed by latex spheres in 93% glycerol at 10 mM KC1. the measured TG = 17 for the doublet Due to the increased "00, the hydrodynamic formed (2). force is now much greater than Fint(h ) and In contrast, the formation and separation the first term in Eq. [1A] predominates. of the orbit pair of short life in aqueous The trajectory constants and values of KC1 shown in Fig, 3 was due to hydrody- hmin computed for 8 doublets are listed in namic forces. The particle trajectories are Table IV where it may be seen that hmin obplotted in Fig. 9. The calculated D and E tained from the paths of recession (column for approach, 1.7 and -0.05 show that the 5), varied from 12 to 25 nm. The paths of spheres were entering a closed orbit for approach and recession were symmetrical which hmin = 350 nm (a high value because when hmin > 15 nm (doublets 13 and 14). By of the large D) and hence Fint(h) = 0 for all contrast, the spheres of doublets 15 to 20, h. After one complete rotation of the doublet whose paths of approach would have taken lasting >10 sec at G = 1.5 sec -1, the par- them to hmin < 10 nm in the absence of inticles separated on a very different trajectory teraction forces, receded on paths having D having D = 3.8 and E = 0.4. Also shown and E larger than the values during approach. plotted in Fig. 9 is the calculated separation However, hmin computed from the recession of sphere centers along the Xa-, vorticity- paths was independent of D and E, being axis which increased from 2b during ap- almost constant at a value of 12 nm, a disproach to 4 b during recession. Translational tance large compared to that, 1/K = 4 nm, at Brownian motion of the spheres probably which double layer interactions are expected accounts for such a drift in relative distance to be important. This is evident in the rightof particle centers and trajectory. hand columns of Table IV which list the d. Transient doublets: hmin < 20 nm. The trajectory constants of the paths of recesabove described trajectory analyses have sion which the spheres would have followed provided strong evidence for the validity of if Eqs. [1], [5], and [6] had applied. The Journal of Colloid and Interface Science. Vol. 82, No. 1, July 1981

186

TAKAMURA, GOLDSMITH, AND MASON TABLE IV Shear-Induced Collisions of 4.0-txm-Diameter PS Latex Spheres: Trajectory Constants in 93% Glycerol; hmin < 30 nm, [KCI] = 10 mM Trajectory constants

Doublet

G (see-1)

13

3.7

14

D~

Calculated recessionpath

approach recession

hmtnb

hmlab

E~

(nm)

0.69 0.69

0.69 0.69

25 25

6.4

0.80 0.80

0.33 0.33

15

4.3

0.46 0.46

16

6.4

17

Dc

recession

Ec

(nm)

0.69

0.69

25

15 15

0.80

0.33

15

0.63 0.65

9 12

0.46

0.63

9

0.48 0.48

0.57 0.62

8 12

0.48

0.57

8

5.3

0.63 0.73

0.06 0.32

2 12

0.65

0.11

3

18

5.5

0.45 0.58

0.06 0.52

0.6 12

0.48

0.20

2

19

3.6

0.20 0.28

0.15 0.80

0.3 15

0.23

0.39

2

20

6.4

0.23 0.35

-0.03 0.73

0.1 12

Formation of permanent doublet

a.b Calculated as in Tables II and III. c From Eqs. [1] using the best fit of the measured path of approach and Fint(h) given by Eqs. [5] and [6] assuming A = 0.003 aJ, k = 200 nm, and tk0 = measured ~-potential. values of D and E, as well as of hmin, are appreciably lower than those computed f r o m the actual paths of recession. DISCUSSION This w o r k has successfully extended the traveling microtube technique to measurements of the trajectories of colliding, charged colloidal-size latex spheres. When the trajectories were analyzed with the aid of hyd r o d y n a m i c theory, previously shown to apply to suspensions of m a c r o s c o p i c spheres (4, 5), it was possible to detect the p r e s e n c e of net attractive or repulsive forces of the order of 0.1 p N (10 -13 N) acting for 0.1 to 0.35 sec while the sphere surfaces approached closer than 100 nm. Thus, the techJournal o f CoUoid and Interface Science, Vol. 82, No. 1, July 1981

nique is about 105× as sensitive as that developed by Israelachvili (7, 8) in which the forces b e t w e e n mica plates are m e a s u r e d directly. It also emerged that, provided h > l/K, the interaction force Fint(h) can be interpreted using the D L V O theory of colloid stability (Eqs. [5] and [6]). Thus, the existence of a net attractive force during collisions of 2.6-/zm-diameter spheres in 10 m M KC1 when 35 < hmin < 80 nm was likely due to the particle passing through the secondary potential energy m i n i m u m w h o s e existence at this KC1 concentration in water had previously b e e n d e m o n s t r a t e d f r o m measurements of the mean TG of p e r m a n e n t doublets (2). The trajectory analysis at this KCI concentration yielded a value of A = 0.003

MICRORHEOLOGY

OF COLLOIDAL

aJ for polystyrene in 50% glycerol, which is reasonable when compared to that, 0.006 < A < 0.01 aJ, obtained in water (2). As previously observed in aqueous 1 mM KC1 (2), no permanent doublets of the 4-/zm-diameter spheres were formed in 50% glycerol containing 1 mM KC1 since a net repulsive force acted during the collisions. This was no doubt due to interactions between the double layers which prevented surfaces from approaching closer than h - 3 5 nm, and which were best described by Eq. [6] using the assumption of constant surface charge density. However, double layer interaction does not appear to be involved in the pronounced repulsion observed in several of the trajectories in 93% aqueous glycerol where the increased hydrodynamic force at a given G resulted in a much closer approach of sphere surfaces, to within h = 12 nm. Relative to the double layer thickness of 4 nm, this represents a long range repulsion for which there appear to be two principal explanations. The first postulates a surface roughness of the spheres which physically prevents approach closer than 10 nm. Such an effect had previously been described for collisions between uncharged, macroscopic 1-mm-diameter spheres in viscous silicone oil when Fint(h ) = 0 (4). Here, the trajectories were found to be symmetrical unless hmin < 0.3 /zm when there was a significant increase in D and E after collision, and the spheres appeared locked together while the doublet rotated between 4)1 = -45 and 0°. Scanning electron micrographs revealed surface asperities of - 0 . 2 /.~m in height. Electron micrographs of the 4 /zm latex were therefore taken in a JEM 7A transmission electron microscope using exposure times < 1 sec to prevent deterioration of particles in the electron beam (27). Measurements of the sphere diameters at a magnification of 3500 x using a Nikon shadowgraph showed that the sphericity of the particles was within ---0.3%, and that at 105× magnifica-

DISPERSIONS

187

tion there was no evidence of any sharp surface asperities or other surface roughness of the order of l0 to 20 nm. This, however, does not necessarily rule out the argument from surface roughness since asperities having thicknesses < 5 nm, sufficient to prevent close approach of spheres, would not have been detected by electron microscopy, nor should the effect of drying in collapsing such surface irregularities be ignored. An alternative explanation may be found in terms of solvation forces, which in water have been reported to act at distances < 6 nm between molecularly smooth surfaces of two mica plates immersed in aqueous electrolyte (8). These forces are also independen~ of the electrolyte concentration. Recently, some theoretical justification for the existence of hydration forces has been provided by two groups (28, 29). Whether these results and theory can be applied to the 93% glycerol-polystyrene system is uncertain, as is the existence of a sufficiently strong solvation force at distances h as large as 10 nm. Whatever the reason for the existence of the repulsion, the results indicate that the DLVO theory, based upon constant surface potential or constant surface charge density, cannot be applied to interactions at h < 10 nm where there is a strong repulsive force acting in addition to that due to double layer interaction. The existence of such a force will prevent the formation of doublets of touching spheres; rather, there will be nontouching primary minimum doublets, in agreement with the previous results (2) that TG for such aggregates > 15.62, the value predicted for dumbbells of touching rigid spheres. It is clear that the traveling microtube technique, together with the trajectory analysis described above, offers a powerful tool for assessing the validity, as well as the limitation of the classical theory of colloid stability. In the following paper, the measJournal of Colloid and Interface Science, VoI. 82, No. 1, July 1981

188

TAKAMURA, GOLDSMITH, AND MASON

urements and analysis are extended to the case where a polyelectrolyte (Cat-floc) is adsorbed onto the particle surfaces (24). NOMENCLATURE

U

X1 (the vorticity) axis in shear flow Electrophoretic mobility of a single latex sphere Fluid velocity along the X3axis (parallel to the tube axis) Cartesian coordinates and axes (i = 1, 2, 3) Permittivity of free space Zeta-potential Suspending medium viscosity Spherical polar coordinates relative to X1 as polar axis Reciprocal Debye double layer thickness London wavelength Surface potential of sphere

U3(R) Hamaker constant; for water, glycerol and polystyrene in vacuo xi, Xi Am Hamaker constant of a mixed solvent e0 A(r*), B(r*), Functions of r* introduced in C(r*) Eq. [1], defined in (5) "0o b Radius of sphere 01, dh D, E Trajectory constant introduced in Eqs. [2] and [3], K defined in (5) Fattr, Frep Attractive and repulsive force h between two spheres tOo Fint Fattr + Frep, total interaction force between two spheres: ACKNOWLEDGMENT attractive (<0) and repulThe authors are grateful to Dr. T. G. M. van de Ven sive (>0) for helpful discussion and advice. f(r*), g(r*) Functions introduced in Eqs. [2] and [3], defined in (5) REFERENCES G; G(R) Shear rate; at R in Poiseuille flow 1. van de Ven, T. G. M., and Mason, S. G., J. Colh, hm~n Gap width between spheres loid Interface Sci. 57, 505 (1976). (r - 2b); minimum value 2. Takamura, K., Goldsmith, H. L., and Mason, S. G., J. Colloid Interface Sci. 72, 385 (1979). during collision 3. Goldsmith, H. L., and Mason, S. G.,in "Rheology: K Dielectric constant of susTheory and Applications" (F. R. Eirich, Ed.), pending medium Vol. 4, p. 86. Academic Press, New York, 1967). 4. Arp, P. A., and Mason, S. G., J. Colloid Interface p Retardation parameter Sci. 61, 44 (1977). = 27rh/k 5. Arp, P. A., and Mason, S. G., J. Colloid Interface Q Volume flow rate Sci. 61, 21 (1977). r,r' Distance between sphere 6. Tabor, D., and Winterton, R. H. S., Proc. Roy. centers, projection on Soc. A312, 435 (1969). X2X3-plane 7. Israelachvili, J. N., and Tabor, D., Proe. Roy. Soe. A331, 19 (1972). r*, r'rain r/b, minimum dimensionless 8. Israelachvili, J. N., and Adams, G. E., J. C. S. value during collision Faraday 1 74, 975 (1978). re(r*) Equivalent spheroidal axis 9. Derjaguin, B. V., Voropayeva, T. N., Kabanov, ratio B. N., and Titiyesvskaya, A. S., J. Colloid Sci. R Radial distance from tube 19, 113 (1964). 10. Barclay, L., and Ottewill, R. H., Spec. Disc. Faraaxis day Soc. 1, 138 (1970). Ro Tube radius 11. Barclay, L., Harrington, A., and Ottewill, R. H., t Time Kolloid-Z.u.Z. Polym. 250, 655 (1972). t* Gt, dimensionless time 12. Homola, A., and Robertson, A. A., J. Colloid InT Period of rotation about the terface Sei. 54, 286 (1976). A; A 11, A22, A33

Journal of Colloid and Interface Science, VoL 82, No. 1, July 1981

MICRORHEOLOGY OF COLLOIDAL DISPERSIONS 13. Derjaguin, B. V., and Landau, L. D., Acta Physicochim. URSS 14, 633 (1941). 14. Verwey, E. G., and Overbeek, J. Th. G., "Theory of the Stability of Lyophobic Colloids." Elsevier, Amsterdam, 1948. 15. Vadas, E. B., Goldsmith, H. L., and Mason, S. G., J. Colloid Interface Sci. 43, 630 (1973). 16. van de Ven, T. G. M., and Mason, S. G., J. Colloid Interface Sci. 57, 517 (1976). 17. Batchelor, G. K., and Green, J. T.,J. FluidMech. 56, 375 (1972). 18. Schenkel, J. M., and Kitchener, J. A., Trans. Faraday Soc. 56, 161 (1960). 19. Frens, G., and Overbeek, J. Th. G., J. Colloid Interface Sci. 38, 376 (1972). 20. Overbeek, J. Th. G., in "Colloid Science" (H. R. Kruyt, Ed.), Vol. 1, p. 333. Elsevier, Amsterdam, 1952.

189

21. Gregory, J., Disc. Faraday Soc. 42, 168 (1966). 22. Churaev, N. V,, Kolloid Zh. 37, 370 (1974). 23. Hiemenz, P. C., in "Principles of Colloid and Surface Chemistry," p. 421. Marcel Dekker, New York, 1977. 24. Takamura, K., Goldsmith, H. L., and Mason, S. G., J. Colloid Interface Sci. 82, 190 (1981). 25. Vincent, B.,J. Colloidlnterface Sci. 42, 270 (1973). 26. Gregory, J., Advan. Colloid Interf~tce Sci. 2, 397 (1969). 27. Vadas, E. B., Cox, R. G., Goldsmith, H. L., and Mason, S. G., J. Colloid Interface Sci. 57, 308 (1976). 28. Marcelja, S., Mitchell, D. J., Ninham, B. W., and Sculley, M. J.,J. C. S. Faraday H 73, 630(1977). 29. van Megen, W., and Snook, I., J. C. S. Faraday H 75, 1095 (1979).

Journal of Colloidand Interface Science, Vol.82, No. 1, July 1981